Question: Solve for $x$ : $ 7|x + 9| - 7 = -1|x + 9| + 8 $
Add $ {1|x + 9|} $ to both sides: $ \begin{eqnarray} 7|x + 9| - 7 &=& -1|x + 9| + 8 \\ \\ { + 1|x + 9|} && { + 1|x + 9|} \\ \\ 8|x + 9| - 7 &=& 8 \end{eqnarray} $ Add ${7}$ to both sides: $ \begin{eqnarray} 8|x + 9| - 7 &=& 8 \\ \\ { + 7} &=& { + 7} \\ \\ 8|x + 9| &=& 15 \end{eqnarray} $ Divide both sides by ${8}$ $ \dfrac{8|x + 9|} {{8}} = \dfrac{15} {{8}} $ Simplify: $ |x + 9| = \dfrac{15}{8}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 9 = -\dfrac{15}{8} $ or $ x + 9 = \dfrac{15}{8} $ Solve for the solution where $x + 9$ is negative: $ x + 9 = -\dfrac{15}{8} $ Subtract ${9}$ from both sides: $ \begin{eqnarray} x + 9 &=& -\dfrac{15}{8} \\ \\ {- 9} && {- 9} \\ \\ x &=& -\dfrac{15}{8} - 9 \end{eqnarray} $ Change the ${ - 9}$ to an equivalent fraction with a denominator of $8$ $ x = - \dfrac{15}{8} {- \dfrac{72}{8}} $ $ x = -\dfrac{87}{8} $ Then calculate the solution where $x + 9$ is positive: $ x + 9 = \dfrac{15}{8} $ Subtract ${9}$ from both sides: $ \begin{eqnarray} x + 9 &=& \dfrac{15}{8} \\ \\ {- 9} && {- 9} \\ \\ x &=& \dfrac{15}{8} - 9 \end{eqnarray} $ Change the ${ - 9}$ to an equivalent fraction with a denominator of $8$ $ x = \dfrac{15}{8} {- \dfrac{72}{8}} $ $ x = -\dfrac{57}{8} $ Thus, the correct answer is $x = -\dfrac{87}{8} $ or $x = -\dfrac{57}{8} $.